Anime Girl Hair And Fiber Bundles

I thought I'd do a small corollary, lecture um concerning um, a nice and important topic in algebraic topology. So, like the Stevenson, Maxwell Paradox and other things like in tamboy theory, this one's gon na be just like one of those um videos that I use to sort of introduce you guys to the notion at least the Notions um or the joys and wonders of algebraic. Topology so um we begin with uh anime girl, hairstyles, so yeah, maybe their bunch of hairstyles that you can think of them would be big deal. So I have a my base. We got. I guess I suck a droids but um, but you can have them on the pigtails. You can have them already slanted. I suppose - and this could be the one - that's um It's like a bought a nice clean, dots now um, I'm Not Really Gon na Focus. Much on the hairstyles hairstyles themselves, but um you will see that we will be leading up to a very nice topic called the tangent bundle. I mean a fiber bundle, so okay. So, let's begin first with how we intuitively think of hair, so we're gon na go a bit into biology here. So if we think of this as the scalp um hair is like some some very very thin thing that has a follicle inside the hair. This is what the hair looks like and there's a small surrounding costing it's the boat. That'S the pore! Now, okay, how does this really the algebraic topology at all? So first, let's take a small cross-section of the hair um which we can think of as a small cylinder, so a small cross-section of hair. If I, if I grab a strand of hair and then just like pull it out and zoom in using a microscope, we can see that it is. You can say if and also be abstract it into our nice mathematical world. We can say that it is homeomorphic to um S1 cross some interval, it's called L. Well, that would be the supposed height of cylinder. So I have a circle S1 and inherently two-dimensional object, and then I take expectation product with um L, which gives me a bunch of these lines. So every point in S1 is associated with um an interval L here, and then each of these would be so if, if this over here is a point, I will call it Alpha. Um each Alpha is paired, so um, no one's very strongly. Okay, Alpha cross L. This one single set here cross L, will give us uh pair or seven oh, a set of bears Alpha and some new um elements that uh such that well, Alpha is now S1 and um. Beta is an element of L. So that's a small sliver of the cylinder so yeah, it's still there, but um in the topology in general. When we talk about cylinders, we're usually talking about the unit cylinder. So it's really the prediction product between S1 and the unit interval zero to one either closed or open um. But if you want to identify, if you really want to identify this or turn the cylinder into a torus, you really want zero and one to be open, or this interval to be open, so um, the set of identifying. You know some some square right and then you identify the points attaching together um before you actually start doing that. Like attaching these points together, you see some. You shouldn't get some identification map right and um the identification yeah. Usually it's. The entire thing is an identification space um. What you want to do is take it open so that when we do identify them both um, there is no ambiguity above which we're going because we're essentially doing some um uh. If I'm, if I'm not mistaken a subjective map - and we want to distinguish these two because when we say that these are this - is a close - a closed interval um, we essentially end up with you know it's sort of um. It is very compact, but to the point we're in it's too compiled. Even I don't think, that's the reason behind it behind the fair explanation for why it must be an open set when we're gon na construct a Taurus out of this cylinder. So, okay, so a strand of hair, even though it's very small, we don't, we don't want to be concerned with the geometric properties off the cylinder or the small silver of air we get from our anime girls um. What we care about is it's General likeness. It'S um, it's General shape. We don't care about this, so much as the dimensions of it, but generally what it looks like locally and globally. Okay, so right, that's one interpretation for what Anatomy girl hair could be. It could be a cylinder, so it could be a cross between some length of L and the unit circle S1 or I don't know it could be some specified circles as long as you identify the atmos. So I have this. I have this um this manifold over here and you can identify or you can specify what exactly the radius is um. So S1 is All Points x, squared plus y Square. All Points x, squared X and Y, such that x, squared plus y squared, is uh equal to some radius, r, squared so X, Y and R is part of an input space R3 um. So so yeah you could have this. You can specify what R exactly is. So if I want a if I want the the radius of my cylinder to be, you know three um three units, so I would say that my specified Circle is of x, squared plus y squared equals 9. x y and that's it are just elements of our R2, I mean you're doing the nine over here is uh, it's a real number, so I don't need to really specify its domain or where it came from. Okay, so um, so in the case the average strand of pair here. So let me just search it up. Real quick is around 40 to 120 micrometers. That'S the diameter so um to get the radius we just sort of have that, so that would be 60, so 60 micrometers! So um. If we say that the average care height of um of girl or meal could be - and you know really so - this also not only applies to girls but um, also to guys um Fanboys um. I suppose your casual anime dilps people like Zorro stuff, like that this applies too or these interpretations, but this lecture we're mostly just going to be focusing on anime girls, okay, so this is one interpretation of it. Oh um, let me just specify okay, so if I have this hair here, the atlas for a manifold. So, if I say h is the hairspray or sh, we could specify using the atlas all points or um this first set and then we take a cross with the length of the hair. So I'm gon na use one of these girls, as my example, l, so um to find the length um. I suppose, if your hair is given by a function or a map, a map from R to R um, it would just have to take um to find its R click. You just need to integrate between two pounds. I believe that's the start and then the end of the interval and then integrate the root of um, one plus this square of the derivative of your function, with respect to X or with respect to the variable using two um, your parameter, basically, okay, so um right. Just a small detour into calculus um, and this is another reason why calcius is actually pretty um important in our study. Is that um yeah so length i l if I have, or if the hair is given Earth. One strand of hair is given by a function. F, that takes R and listen to R um. The legs of f would be some integral from into b a b minus a is precisely the abscissa, I believe, or the length between um one, our first parameter and the second parameter. So a would be n b weave input into the final anti-derivative in order to get the real length of our you know, hair strap and that's the square root of one plus the square of the derivative with respect to our parameter t of or X, which is X, um of f uh, I'm sorry squared so we're not taking the second derivative, but just the square of the derivative, okay and then just integrate that with respect to X. So we take a small sliver. Okay, and this slice here is actually like. You can division as a triangle. This is the hair strand, we're taking a small slice or sliver of it and, as we get closer closer because this DX here in place we're making it infinitely thin, we eventually arrive with you know the actual length of a small sliver of that hair, okay or At least the sun, because when we use this, we're implying right, like a very, very fine sum, an infinitely fine Sun of segments, so yeah or if I want to parametrize the the function in with respect to some parameter T, I would suppose we'd use the chain. Rule here and imply, of course, that would be DT and that would be p x, f d x of t with respect to T: okay, yeah, okay enough by calculus. Let'S return back to um our our subject. How about I have a tissue, okay, um! So, okay yeah this stuff, like that pretty cool stuff that works okay, uh well, this is a little inconvenient. Excuse me, so I'm just gon na pause this lecture and then once I get some tissue, we're gon na clear up the board and then continue on okay. Um, I made a half issue, but this rag will do let's erase that and specify our um are manifold. So it's technically a manifold okay, so our sh it let's go to um. I have point x: square plus y squared is equal to some um squared I'll specify the unit here. U m, squared such that um such that x y are in R2, oh and cross the length l of our hair. It was a set, so I'm gon na use I'm gon na get stupid, so there it is so this is the this is the entire map, or this is the Atlas we used to describe our our manifold, okay, so locally speaking, if we zoom in it would Look like a two-dimensional slice um or it would look two-dimensional to us, but it's really three-dimensional globally, speaking, Okay, so right! So what? What would be the fundamental group of this thing so another crucial Topic in other topology fundamental group of a space? Now because we say we say it's like it's gon na crack, okay, we say space is contractable to a point. If we can map all points in that space onto that single point, so we can say that R2 or RN is contractable to the origin, which is zero okay. So now this is one way for us to find the fundamental group of a space, and the fundamental group of a space is a is a group mathematical structure that gives us it gives. It definitively tells us what the group is and how we can um categorize. It so we're categorizing spaces right um. You guys will see in the course about categorizing, F, spaces and stuff like that or fembois faces um, and then we're essentially saying if two space are definitively different from each other right. So we want to say: if um, let's say I poke a hole, I grab a needle and that's very thin and poke a hole through a um through a cylinder and then okay, you want to identify, if that, one over there and another piece of hair that I didn't poke are fundamentally different right yeah. I know you're, probably asking me why the hell would you want to torture your poor anime girls this way, but in reality, if I poke your a piece of strand of your hair, it really would hurt okay or hell. If I, if I were to stab an anime girl in her chest um, how would I tell she is fundamentally different from an anime girl that I did not stab in the chest right so yeah? That'S one way of use of of of determining that by finding the fundamental group. So, okay, going back. For example, if I were to poke the hole through a strand of hair. So I have a sliver of the strand of hair and I would have poke a hole through it or um. Speaking in terms of topology take away a disc. I take away like this. I take away this from it or two discs or a neighborhood or a small cylinder part of it right, because this strand of here is um is a is a solid cylinder. Okay, it's a solid cylinder, there's something inside it, so we essentially think about support. Now. I know what you're thinking the 100. This is it I described a while ago. It'S not really a solid cylinder. What we just have to do to say that this is a solid cylinder, is just it said, say that we're taking the union or not the union, but the Cartesian product between a disc and and the discs and our length of hair or length of the leg Of the cylinder we want all right, so this is a disc and the address will just be well x. Squared was y squared, but instead we say it is less than r squared times y r in R3 um. This is for the Open disc. The closed. This would just say it is also equal, so it could be equal to r squared, and so it includes the boundary also, which is just the uh. The circle S1 so S1 is not really a solid figure. It is just like a ring of points. Well, not not really ring but like ring points, okay, so fundamental groups. So what's the formula group of a cylinder now cylinder, a cylinder is just a bunch of of um this sap on top of each other, and this is a concave subset of r two. Now, concave subsets of R and according to a Lima that we will discuss later on the course have fundamental group of zero or yeah zero, and because of that, we have a formal group of zero here. Also, but the thing with um, a piece of hair, a strand of anime girl, hair that I poke um I poked through, is that it has a missing subset in it. So there is part of it that isn't concave it's open and then I could run a string through it. It would be inside the set, but um there will be parts of it that are sort of inside, but not really inside. So it's not con. It'S not convex anymore. Sorry, I meant convex, not concave, so it's a concave shape now. So therefore it wouldn't have upon my group of zero. It would have some other fundament group, okay and in fact, if I were to glue the point, so I have great. I have my thing that I poked through I have a hole inside it and then I zip that whole thought it would give me a Taurus, and the fundamental group of a Taurus is not really zero. It is the integer Z. The former group of this year is just zero, so this is no by pi zero and if I poke a hole through here poke a hole through this, you essentially see that okay um with a little stretch of imagination, you can round out these edges to get A and right we have a Taurus and where is it in the edges of this um, this hole and the fundamental group of a Taurus? Oh wait. Sorry, this is not so buy. One of a normal cylinder gives us zero, but okay, so there is a homeomorphism or homomorphism between the final group of this and Obama group for this. But there is no homomorphism between this one in this formula group, because these two space are not homeomorphic to each other. We cannot morph this into here without taking a subset away, so the polymer group of this is actually Z, integers, but Z is not precisely zero. So we can say with confidence that these two spaces are not the same, and thus these two spaces are not the same: okay, okay, so, okay, um, disregarding that, let's move on to the actual meat of what I want to discuss today, and that is the fiber. What is a fiber, you can think of a fiber which is it's a hair strand, it's piecing um. It could be a rope and then you um, you intertwine the ropes as a fiber like in the physical sense, but in the mathematical sense a fiber is a much bigger and better construct. It is an element of what is called a fiber bundle and to think or to visualize the Bible. This is exactly why I use the analogy of an anime, girl's, hair or scalp, because the scalp can be thought of as the base space. The hair can be done as the fibers and the ambient air around her everything around her, including the hair strands, is the total space e. So a fiber bundle definition. Fiber bundle is a structure or it could be a topological space. It'S most definitely topological space because it has topological spaces in it. So it's a structure which is e B pi and F such that by it Maps. Um e up to the base space satisfies the homophobia lifting properly so this map, Pi, is not a the fundamental group thing, but it is a different map. At this point, um in fact Pi now is called the um pie is called. It is called the the projection map and it is a continuous surjection, so it makes sense because these are Total. Space includes the fiber and every other point around it. So when we're mapping those points onto B, we're essentially mapping more points onto one point, because what this essentially does is map the fibers or any point on the fibers onto the roots in the base space. That'S what it does so. Okay, I have the scalp. This is our base space b, which is the scalp of the road. It'S it's a very it's like like how we zoom in on a ball and think it's pretty much flat. This one here is also flat. I just think it's also flat he's infinitely thin and I have the hair, oh hair, which are the fibers. So this is f and then e is the five original and then what pi does is take any point on these fibers and map them onto the base. They map them onto the base, and so okay. So now I'm gon na go in depth exactly on now. These fibers are not precisely cylinders, but these guys are just like literally sets of points. That'S um that satisf. That also goes along with. Why the satisfied the the next or the property, I'm gon na the property, I'm gon na, explain and so um. It'S a triviality condition that by most satisfy so okay. So all right! Um with this, I'm also going to assume that b, our base space for the scalp of the anime girl is compact. Okay and the condition is that, for every Point, Beach inside B, every poetry inside B, there is an open neighborhood. There is this an open neighborhood View you off B um, which we call a trivializing neighborhood such that uh such that there is a homeomorphism. There exists a homeomorphism which I would call Phi that takes the inverse um or the image of this open neighborhood and takes it to a new cylinder which is um. You partition product f. So it takes it to this partition point. So it's a cylinder cup of cylinder um we're. Obviously this here uh implies the Subspace topology. We need the Subspace topology for this to work, and this year implies the product topology. So, every time you take things like products, we construct subsets from a set um or we take images of or pre-images of a subset of a topological space. We need to apply a Subspace topology or a product topology or if we were to apply a metric over a Subspace or a space itself. It must imply also not only the regular or standard the body, but either the euclidian topology or a metric topology over that space. Okay. So now I have this map here, um uh, so I have this map such that the following um diagram, commutes, so um. This is another thing that you guys will encounter. A lot in this course is that if you decide to go into it, obviously um I'm making this video, while I'm in the middle of making the course so yeah such that the following diagrams, you guys will be encountering commutative diagrams a lot, especially when we're going To start um discussing about implied homomorphisms from the fundamental groups constructed from topological spaces, um so yeah, so this diagrams so by the minus one of you open, neighborhood. Of course it's not using this you're just gon na um it just sort of showing the relation. Then this maps onto just under Pi, just you, so we just compose Pi with by to get to you. Obviously, but you know this inverse wouldn't exist. If you know you wouldn't have some in inverse where, if we compulsive, we will just get the identity. Let'S do nothing okay and then we get a projection now so right we have a cylinder over here. You can start out of the Subspace. U and then F, which is basically the fibers um. So this is another reason why I discussed through this, because it gives us an intuition for how this this mapping works. So I have you or strange little subset of the scalp of our anime girl and I have some hair. So it gives us a sort of um sort of cylinder like object, an object, that's homomorphic or cylinder, and then I have a projection map that takes all of these points and just squashes them out down to you. So I have a point here and then on the projection sub 1. It takes it it corresponding point over here and um. This would also be this would be a surjection. I may need to one mapping because we're squashing a bunch of points we're taking a bunch of points and mapping them down to you know um one batch of points. Okay, so this projection, subword, is called the natural projection, and this Phi over here is a homeomorphism. So the set of all you know: um subsets indexed by some, I equipped with this map over here. It'S called the local tribalization of our bundled or our fiber bundle. So this is the condition that are not. You know. Five, I mean High Five by most satisfying the trivialization stuff, so we can construct a cylinder out of U and F our fungal r fibers and then using have a bunch of maps from you know. The free image of you under you know our transformation um getting map onto our cylinder and then you can take its in inverse and get your step open, neighborhood, and then we have projection that takes this cylinder and then Maps it on to just that. Opening about peace, okay, so that is a fiber bundle and that's another good and that's it's a very important tool in algebraic, topology um. It'S not really a tool, but it's an object. Really. Nice object in algebraic apology, that's so habits to connect a lot with cylinders, and you know um with that connect somewhat to anime girl. You know hair or any hair it could be, it could even be a human being, just a regular real person or just it could be a guy also, basically care in general, and actually the the analogy that you're given when you first learn about this is a Hairbrush, how fitting right so the bristles of the hair brush? You can also think of a toothbrush. The bristles of it are the fibers and then the the main thing itself is the the base space and everything else is the E portal space. So, going back to five, what pi essentially does is take a steak any you know um any point in F and then takes it to the root or, if I go back to the scalp, an example of a scalp to draw my flat plane, which is the Scalp or anime girl, and then I have the fibers. So I have a point here. I will call it Q. It takes it down to a root here. Pipe takes you down a route over there or a point where, in the fiber and the base space meet as actually fiber. Bundles are a really important thing when we start discussing about half vibrations and very crazy. If you go to half link, which is a basically two two circles linked together, then this one is supposed to be inside someone inside. So this is over. This is a half link and then hot vibration usually happens when we have do we take its surface. So a half surface now and the half vibration is basically a visualization or projection of a fourth dimensional sphere or S3 onto S2 or onto a um, the two sphere, basically so the three sphere to the two sphere. So I have the surface here, which is the hot surface. I know it's a bit hard to visualize, especially since I don't have a colored marker to tell which one is outward, which is which part of the surface is inward, so yeah just sort of bare weight. Yeah, so fiber bundles are going to be very important when visualizing a are starting to discuss about off vibrations um, because mostly because um, not only because of the name, you know vibrations or have relations to basically fiber bundles. Here um yeah, like we can think of every surface possible. Even the base space itself can be thought of as a um as a fiber bundle just take an infinite amount of fibers. So the base phase of the fiber bundle, which we think of as the base space, would be this line over here. This interval over here, and then we have another one over here. So it's an infinite amount of fibers, packed it together that we can think of any service like that. So things like the Mobius band, um, even a regular cylinder, an annulus we can think of them as fiber bottles um. But the thing with the Mobius band is that it's not orientable uh, it's a non-orientable space. It'S non-trivial: it's a non-trivial, even though it looks easy to construct it's a non-trivial fiber bundle, because because when we identify the the sides of our identification space, we have to twist it 180 degrees before connecting them together. Okay, so that is the fiber bundle and its correlation with anime girl, hairstyles, I'm not really a hairstyles, but just the hair itself, and just how connected to the scalp and stuff like that, I'm going to do a little bit of biology and, frankly, um. I'M happy what this lecture came last so hope you guys enjoy algebraic family Theory, and I hope this video at least give you guys some motivation on trying to learn, and you know and homologies so yeah. That'S a good place to stop.